Theorem clnbgrssvtx 47756

Description: The closed neighborhood of a vertex 𝐾 in a graph is a subset of all vertices of the graph. (Contributed by AV, 9-May-2025.)

Hypothesis

Ref	Expression
clnbgrvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
clnbgrssvtx	⊢ (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉

Proof of Theorem clnbgrssvtx

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clnbgrvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	clnbgrisvtx 47755	. 2 ⊢ (𝑛 ∈ (𝐺 ClNeighbVtx 𝐾) → 𝑛 ∈ 𝑉)
3	2	ssriv 3999	1 ⊢ (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉

Syntax hints:   = wceq 1537   ⊆ wss 3963  ‘cfv 6563  (class class class)co 7431  Vtxcvtx 29028   ClNeighbVtx cclnbgr 47743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-clnbgr 47744

This theorem is referenced by:  clnbgrlevtx  47769  clnbgrisubgrgrim  47838  clnbgrgrim  47840  isubgr3stgrlem6  47874  isubgr3stgrlem7  47875  isubgr3stgrlem8  47876  isubgr3stgr  47878  uhgrimgrlim  47890  grlicref  47908  grlicsym  47909  clnbgr3stgrgrlic  47915